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ECON 3210/4210 Decisions, Markets and Incentives Lecture notes 2.11.05 Nils-Henrik von der Fehr STATIC GAMES OF INCOMPLETE INFORMATION Introduction Complete information: payoff functions are common knowledge Incomplete information: at least one player is uncertain about another player’s payoff function Bayesian games Examples (sealed-bid) auctions bargaining (dynamic) Representation normal form or extensive form beliefs Solutions: Refinement of Nash Equilibrium Bayesian Nash Equilibrium beliefs at equilibrium Normal-Form Representation The normal-form representation of an n-player static Bayesian game specifies the players’ strategy (action) spaces S1,...,Sn ( A1,..., An ), their type spaces T1,...,Tn , their beliefs p1,..., pn and their payoff functions u1,..., un . The game is denoted G = {S1,..., Sn ;T1,...,Tn ; p1,..., pn ;u1,...,un } . Player i’s type, t i , is privately known by player i, determines player i’s payoff function ui ( s1,..., sn ; t i ) and is a member of the set of possible types, Ti . Example high-cost versus low-cost firms successful versus unsuccessful developers of new technology Player i’s belief pi ( t − i t i ) describes i’s uncertainty about the n-1 other players’ possible types, t − i , given i’s own type, ti . The function pi ( t − i t i ) gives the probability (as seen by player i) that opponents’ types are t − i , conditional on player i’s type being ti . Example probability distribution over competitors’ costs probability distribution over whether competitors have succeeded Note: in both cases beliefs may depend on own type (eg. more likely that competitor has low costs (has been successful), if the firm itself has low costs (has been successful)). Problem: How are beliefs determined? How do we ensure that beliefs are compatible? Timing (Harsanyi, 1967) 1) Nature draws a type vector t = ( t1,..., tn ) , where ti is drawn from the set Ti . 2) Nature reveals t i to player i, but not to any other player. 3) Players simultaneously choose strategies (actions). 4) Payoffs are received. Note that this set up transforms a game of incomplete information into a game of imperfect information – the move of Nature is not (perfectly) observed and hence players do not know the whole history of the game when they have to make their move. It is assumed that the probability distribution according to which Nature moves is common knowledge; that is, all players know that Nature draws t = ( t1,..., t n ) according to the probability distribution p ( t ) . 2 After individual types have been revealed, individual beliefs can be calculated using Bayes’ rule: pi ( t − i t i ) = p ( t − i , ti ) p ( ti ) = p ( t − i , ti ) ∑ p (t ) i t − i ∈T− i where p ( t − i , t i ) is the probability that nature draws type t i for player i and types t − i for i’s opponents, and p ( t i ) is the (marginal) probability that nature draws type ti for player i. It follows that other players can compute the various beliefs that player i may hold, depending upon their types. Example: Two players, 1 and 2, play a game. Player 2 may be of one of two types, t = −1 or t = 1 . The probability that Player 2 is of type t = −1 is p. Given Player 2’s type, the game may be described by the following table: Player 2 Player 1 L R U 1,t -2,0 M 0,2 0,2 D -2,0 1,-t The normal-form representation may be described as follows: Players’ strategy spaces are S1 = {U, M, D} and S2 = {L, R} . Player 1’s type set is a singleton (the type is common knowledge), while Player 2’s type-space is T2 = {−1,1} . The beliefs of Player 1 about Player 2’s type is given by Pr ( t = −1) = p and Pr ( t = −1) = 1 − p . The payoffs are as described in the table above. Extension: games in which players have private information not only about their own payoffs, but about other players’ payoffs also (example: asymmetric information about demand conditions in oligopoly). Consequently, payoffs are functions of other player’s types also, u ( s1,...sn ; t1,..., t n ) . 3 Bayesian Nash Equilibrium In a static Bayesian game G = {S1,..., Sn ;T1,...,Tn ; p1,..., pn ;u1,...,un } , a strategy for player i is a function si ( t i ) , where for each type t i in Ti , si ( t i ) specifies the action from the feasible set Si that type t i would choose if drawn by nature. In a separating strategy, each type chooses a different strategy, i.e. si tˆi ≠ si ti if tˆi ≠ tj . In a pooling strategy, all types choose the same ( ) ( ) strategy, i.e. si ( t i ) = si , all t i ∈ Ti . In the static Bayesian game G = {S1,..., Sn ;T1,...,Tn ; p1,..., pn ;u1,...,un } the strategies s * = ( s1 *,...sn * ) are a (pure-strategy) Bayesian Nash equilibrium if for each player i and for each of i’s types t i , si * ( t i ) solves max si ∑ p (t i −i t i ) ui ( s1 * ( t1 ) ,..., si ,..., sn * ( tn ) ; t i ) t− i In other words, given a player’s belief about the other players’ types, his or her strategy is a best response to the other players’ strategies. This is true whatever type player i may be. Example continued: In the above example, Player 2 has a (weakly) dominant strategy: if his type is t = −1 the best response is R whatever the choice of Player 1; if his type is t = 1 the best response is L. Suppose Player 1 believes that Player 2 will play L when he is of type t = −1 – which occurs with probability p – and play R when he is of type t = 1 – which occurs with probability 1-p. Then Player 1’s payoff from playing his various strategies are U : [1 − p ] ⋅ 1 + p ⋅ [ −2] = 1 − 3 p M : [1 − p ] ⋅ 0 + p ⋅ 0 = 0 D : [1 − p ] ⋅ [ −2] + p ⋅ 1 = 3 p − 2 It follows that U is a best response if p ≤ 31 , D is a best response if p ≥ 32 , while M is a best response if 31 ≤ p ≤ 32 . Consequently, we have the following equilibria 4 1 : s1 * = U, s2 * ( −1) = R, s2 * (1) = L 3 1 2 < p < : s1 * = M, s2 * ( −1) = R, s2 * (1) = L 3 3 2 p > : s1 * = D, s2 * ( −1) = R, s2 * (1) = L 3 p< When p = 31 and p = 32 we have multiple (i.e. two) equilibria. (Note that, in this game, Player 2 may actually benefit from the fact that Player 1 does not know his type; hence he has an incentive to hide his true type, i.e. to make p close to 0.5.) First-Price, Sealed-Bid Auction Suppose two bidders participate in an auction for one object. Bidder i has valuation (i.e. maximum willingness to pay) v i for the object, i = 1,2. Bidders’ valuations are independently and uniformly distributed on [0,1] . Bidders are risk neutral. Bidders simultaneously submit their bids, which must be nonnegative. The bid of player i is denoted bi , i = 1,2. The bidder with the highest bid wins the object and pays a price equal to his bid (if bidders submit the same bid, the object is allocated on a 50-50 basis). The loser gets and pays nothing. In this game, player i’s strategy space is Si = [0, ∞ ) , while the type space is Ti = [0,1] . Because valuations are independently drawn, player i’s belief is that v j is uniformly distributed on [0,1] , no matter what the value of v i is. Player i’s payoff is ⎧v i − bi if bi > b j ⎪1 ui ( b1, b2 ;v1,v 2 ) = ⎨ 2 [v i − bi ] if bi = b j ⎪0 if bi < b j ⎩ Suppose bidder i believes bidder j’s strategy – the function from type to action – is given by b j (v j ) . Let Fj ( b ) denote the probability that bidder j bids below b. Then bidder i chooses his bid so as to solve the problem max bi [v i − bi ] Fj ( bi ) . The first-order condition for this problem may be written 5 −Fj ( bi ) + [v i − bi ] Fj′ ( bi ) = 0 . Note the probability that bidder j bids below b is equal to the probability that bidder j’s valuation is below v, where v is such that b j (v ) = b . Since valuations are uniformly distributed on [0,1] , it follows that Fj ( b j (v ) ) = v . By implicit derivation, we find Fj′ ( b j (v ) ) b′j (v ) = 1 or Fj′ ( b j (v ) ) = 1 b′j (v ) . We are looking for a symmetric equilibrium, in which both players follow the same strategy; that is, b1 (v ) = b2 (v ) = b (v ) and F1 ( b ) = F2 ( b ) = F ( b ) . Substituting into the above equations and solving, we find −v + ⎡⎣v − b (v ) ⎤⎦ 1 =0, b (v ) which may alternatively be written b′ ( v ) v + b ( v ) = v . Note that the left-hand side is equal to d dv ( b (v ) v ) . Then, integrating both sides with respect to v, we find b (v ) v = 1 2 v +A 2 Note that the bidder type with the lowest valuation must bid zero; that is, b ( 0 ) = 0 . It follows that b (v ) = 1 v 2 So, at equilibrium each bidder submits a bid equal to half his valuation. With n bidders, one can show that strategies are given by b (v ) = n −1 v, n which implies that bidders bid more aggressively (i.e. closer to their valuations) the more bidders there are. The Revelation Principle See Gibbons, ch. 3.3. 6